1) Determine the amount of heat needed to turn 500g of ice in a 100g aluminum pan into steam. My answer is 360,000 cal

If you are going to specify that we have an aluminum pan then we need to know what temperature the pan is starting at. Additionally we need to know what temperature the ice starts at as well. There is not enough information present to do this problem.

The maximum amount of heat energy the water can have is when its initial temperature is at the boiling point, 373 K. (Otherwise it is steam, not water.) Assume the copper pot also has this temperature at the beginning.

Let's find out how much heat needs to be absorbed from the system to turn the water at the boiling point into ice:
(The copper pot must be a maximum of 273 K for the water to freeze, so I'm assuming that it drops at the same temperature rate as the water. This is not quite physically true, but will do as an approximation.) The first two terms take care of the heat needed to be absorbed to take the water and copper from 100 C to 0 C, the third term is the heat to be absorbed from the water at 0 C to turn it into ice.

The coolant absorbs of heat energy. This is less than the heat we can absorb from the water and copper at 100 C to freeze the ice, so we would need a starting temperature for the water-ice system in order to give a definitive answer to this question.

(Now if the coolant absorbed, say, then we would know for sure that the water would freeze.)

If you are going to specify that we have an aluminum pan then we need to know what temperature the pan is starting at. Additionally we need to know what temperature the ice starts at as well. There is not enough information present to do this problem.